Calculations over input variables can be used to produce one or more results, and the results typically have an associated error. This error can be due to various factors, such as imprecise measurement, rounding, etc. For example, if a calculation produces a dollar amount rounded to the nearest penny, then this rounding contributes to an error of ±$0.005 cents, due to the fact that a calculated result such as $16.72 could actually represent any amount between $16.715 and $16.725.
Some calculations are based on alternatives of different calculations. For example, consider two calculations represented by the two functions ƒ(x) and g(x). A third calculation, h(x), could be based on the rules h(x)=ƒ(x), if x≦10, and h(x)=g(x), if x>10. If ƒ(x) and g(x) each have associated errors, εa and εb, respectively, then h(x) inherits error from functions ƒ and g. However, h(x) also has a third potential source of error, based on the uncertainty as to whether h(x) will, in a given case, be determined by ƒ(x) or g(x). When x is very near the boundary between where ƒ applies and where g applies (e.g., where x≈10), uncertainty about the exact value of x can also create uncertainty as to whether ƒ or g applies, and this uncertainty is a third source of error apart from εa and εb. One may wish to take this third source of error into account.